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QUADRATICS
INTRODUCTION TO QUADRATICS
Algebra 2
Real Life: Why would we ever need this?
■ Quadratics have a highest power of 2
■ Standard Form
• f(x) = ax2 + bx + c
• a, b, and c are constants
What Makes a Quadratic a Quadratic?
■ There are 4 ways to solve quadratic equations:
1. Taking Square Roots
2. Factoring
3. Quadratic Formula
4. Completing the Square
Ways to Solve Quadratics
■ When do we use this…..
– Whenever b is 0
– In other words, when you only have an x2
Solving Quadratics: Taking Square Roots
■ n2 = 16 ■ x2 = 80
Solving Quadratics: Taking Square Roots Examples
■ -3k2 = -132 ■ 9v2 = 531
Solving Quadratic Equations: Taking Square Root Examples
■ 8b2 + 7 = 15 ■ 5n2 + 6 = 91
Solving Quadratic Equations: Taking Square Root Examples
■ -10 – 9b2 = -874 ■ 4x2 + 9 = 41
Solving Quadratic Equations: Taking Square Root Examples
■ Taking Square Roots
Solving Quadratics: Taking Square Roots Homework
Bell Ringer
Solve the following quadratic equations.1. 8x2 -5 = 3
2. 6x2 + 7 = 277
FACTORING INTRODUCTION
Algebra 2
Ways to Solve Quadratics
■ What are the 4 ways to solve quadratics?
1. Taking Square Roots
2. Factoring
3. Quadratic Formula
4. Completing the Square
Solving Quadratics By Factoring
What is a factor?
• a number or quantity that when multiplied with another
produces a given number or expression
• Example: 2 is a factor of 12, 7 is a factor of 63
When do we factor quadratic equations?
■ When a is 1
Solving Quadratics by Factoring: Examples
■ x2 – 10x + 24■ x2 + 14x + 48
Solving Quadratics by Factoring: Examples
■ x2 + 11x + 30■ x2 + 2x – 15
Solving Quadratics by Factoring: Examples
■ x2 – 2x – 3 ■ x2 – 2x – 35
Solving Quadratics by Factoring: Homework
Solving Quadratics by Factoring Day
1
Bell Ringer
■ Factor each of the following quadratic expressions
1. x2 – 11x + 28
2. x2 + x - 30
SOLVING QUADRATIC EQUATIONS BY
FACTORINGAlgebra 2
Ways to Solve Quadratics
■ What are the 4 ways to solve quadratics?
1. Taking Square Roots
2. Factoring
3. Quadratic Formula
4. Completing the Square
Solving Quadratics By Factoring
What is a factor?
• a number or quantity that when multiplied with another produces a given
number or expression
• Example: 2 is a factor of 12, 7 is a factor of 63
When do we factor quadratic equations?
■ Easiest and most recognizable when a is
1
■ Will focus on when a isn’t 1
Solving Quadratic Equations by Factoring: Examples
■ Find the zeroes of the following quadratic equation.
x2 – 6x + 5 = 0
Solving Quadratic Equations by Factoring: Examples
■ Find the zeroes of the following quadratic equation.
x2 – x – 42 = 0
Solving Quadratic Equations by Factoring: Examples
■ Find the zeroes of the following quadratic equation.
3𝑝2 − 2𝑝 − 5 = 0
Solving Quadratic Equations by Factoring: Examples
■ Find the zeroes of the following quadratic equation.
3𝑛2 − 8𝑛 + 4 = 0
Solving Quadratic Equations by Factoring: Examples
■ Find the zeroes of the following quadratic equation.
5𝑛2 + 19𝑛 + 12 = 0
Solving Quadratic Equations by Factoring: Examples
■ Find the zeroes of the following quadratic equation.
9𝑘2 + 66𝑘 + 21 = 0
Solving Quadratic Equations by Factoring: Homework
Solving Quadratic Equations by
Factoring
Bell Ringer
Solve the following quadratic equations by factoring
1. 35x2 - 17x – 30 = 0
2. 6x2 + 49x + 8 = 0
SOLVING QUADRATIC EQUATIONS WITH THE QUADRATIC FORMULA
Algebra 2
Ways to Solve Quadratic Equations
■ What are the 4 ways to solve quadratic equations?
1. Taking Square Roots
2. Factoring
3. Quadratic Formula
4. Completing the Square
When do we use the Quadratic Formula?
■ Can always use the quadratic formula, but
it isn’t always the easiest method…
Solving Quadratic Equations using the Quadratic Formula
■ What is the standard form of a quadratic equation?
■ What is the quadratic formula?
Solving Quadratic Equations using the Quadratic Formula: Examples
■ 4x2 + 5x – 26 = 0 ■ 2r2 – 3r + 9 = 0
Solving Quadratic Equations using the Quadratic Formula: Examples
■ 4x2 + 6x + 8 = 0 ■ 5x2 + 8x + 16 = 0
Solving Quadratic Equations using the Quadratic Formula: Examples
■ 4x2 – 5x + 4 = 0 ■ 6x2 – 4x + 6 = 0
Solving Quadratic Equations using the Quadratic Formula: Homework
Solving Quad Equations: Quad Formula
Day 1
Bell Ringer
Solve the following quadratic equations using the
quadratic formula
1. 2x2 - 7x – 9 = 0
2. 3x2 + 2x + 11 = 0
SOLVING QUADRATIC EQUATIONS WITH THE
QUADRATIC FORMULA: DAY 2
Algebra 2
Solving Quadratic Equations with Quadratic Formula: Examples
■ 6x2 + 8x – 73 = -9 ■ x2 – 12x – 11 = -10
Solving Quadratic Equations with Quadratic Formula: Examples
■ 9x2 + 2x = –12 ■ 2x2 + 4 = –5
Solving Quadratic Equations with Quadratic Formula: Homework
Solving Quad Equations: Quad Formula
Day 2
Bell Ringer
Simplify the following radicals:1. 392
2. −98
3. −128
4. 112
SOLVING QUADRATICS BY
COMPLETING THE SQUARE
Algebra 2
4 Ways to Solve Quadratics
The 4 ways to solve quadratics are:
1. Taking Square Roots
2. Factoring
3. Quadratic Formula
4. Completing the Square
Solving Quadratics By completing the Square: When?
■ When do we complete the square to solve
quadratics:
– When b is even and a is 1
Solving Quadratics By Completing the Square: Steps
Steps to Completing the Square
1. Move all constants to the right (away from the letters)
2. Take b and cut it in half, then square that number
3. Add this number to both sides
4. Factor the left
5. Solve
Solving Quads By Completing the Square: Examples
■ x2 + 2x – 99 = 0
Solving Quads By Completing the Square: Examples
■ x2 + 4x – 30 = 0
Solving Quads By Completing the Square: Examples
■ x2 + 20x – 104 = –9
Solving Quads By Completing the Square: Examples
■ x2 – 18x – 43 = –6
Solving Quads By Completing the Square: Examples
■ x2 – 2x – 89 = 8
Solving Quads by Completing the Square: Homework
■ Solving Quads: Completing the Square
Worksheet
Bell Ringer – Completing the Square
𝑥2 + 30𝑥 − 9 = 0 𝑥2 + 14𝑥 + 84 = −14
Completing the Square
𝑛2 − 16𝑛 + 60 = 0 𝑏2 − 2𝑏 − 80 = 0
Completing the Square
𝑚2 + 2𝑚 − 8 = 0 𝑥2 + 20𝑥 − 37 = 0
Completing the Square
𝑥2 − 2𝑥 − 3 = 0 𝑥2 + 18𝑥 + 72 = 0
Completing the Square
𝑥2 − 12𝑥 − 28 = 0 𝑚2 − 8𝑚 + 7 = 0
Completing the Square
𝑝2 − 6𝑝 − 17 = 0 𝑚2 + 8𝑚 − 88 = 0
Completing the Square
𝑏2 − 20𝑏 − 71 = 6 𝑎2 − 2𝑎 − 36 = 6
Completing the Square
𝑝2 − 4𝑝 − 62 = 2 𝑝2 + 6𝑝 + 3 = 10
Completing the Square
𝑥2 − 8𝑥 − 42 = 9 𝑥2 − 10𝑥 − 94 = 9
QUADRATIC WORD PROBLEMS
Algebra 2
Solving Word Problems
■ Underline important info
■ Circle verbs/what being asked to do
■ Always ask yourself: What do they really want me to find?
Shortcut to Find Vertex
■ To find the x-coordinate of the vertex
• 𝑥 = −𝑏
2𝑎
– How would you find the y-coordinate of the vertex?
– Plug in the x-coordinate found
Word Problems
The height, h, in feet of an object above the ground is given by h = -16t2 + 64t + 190,
where t is the time in seconds.
a. Find the time it takes the object to reach its maximum height.
b. Find the maximum height of the object.
c. Find the time it takes the object to hit the ground.
Word Problems
■ The path of a rocket is given by the equation: ℎ = −16𝑡2 + 128𝑡, where h is the
height in feet of the rocket and t is the time in seconds after it is launched.
a. How long is the rocket in the air?
b. What is the maximum height the rocket reaches?
c. About how high is the rocket after 1 second?
Word Problems
A manufacturer of tennis balls has a daily cost of C(x) = 200 – 10x + 0.01x2, where
C is the total cost in dollars and x is the number of tennis balls produced. What
number of tennis balls will produce the minimum?
Word Problems
The value of Jon’s stock portfolio is given by the function v(t) = 50 + 77t+ 3t2,
where v is the value of the portfolio in hundreds of dollars and t is the time in
months. How much money did Jon start with? What is the minimum value of Jon’s
portfolio?
Word Problems
Find the number of units that produce the maximum revenue, R = 900x − 0.1x2,
where R is the total revenue (in dollars) and x is the number of units sold.
QUADRATIC GRAPHS AND THEIR PROPERTIES
Vocabulary■ A quadratic function is a type of nonlinear function that models certain
situations where the rate of change is not constant.
■ The graph of a quadratic function is a symmetric curve with the highest
or lowest point corresponding to a maximum or minimum value.
■ Standard Form of a Quadratic Function:
– A quadratic function is a function that can be written in the form
𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐, where a ≠ 0. This form is called the standard
form of a quadratic function.
– Examples: 𝑦 = 3𝑥2 𝑦 = 𝑥2 + 9 𝑦 = 𝑥2 − 𝑥 − 2
Vocabulary
■ The simplest quadratic function 𝑓 𝑥 = 𝑥2 𝑜𝑟 𝑦 = 𝑥2 is the quadratic parent function.
■ The graph of a quadratic function is a U-shaped curve called a parabola.
■ You can fold a parabola so that the two sides match exactly. This property is called symmetry.
■ The fold or line that divides the parabola into two matching halves is called the axis of symmetry.
Vocabulary
■ The highest or lowest point of a parabola is its vertex, what is on
the axis of symmetry.
■ If a > 0 in y = ax2 + bx + c, the parabola opens upward. The vertex
is the minimum point, or lowest point, of the parabola.
■ If a < 0 in y = ax2 + bx + c , the parabola opens downward. The
vertex is the maximum point, or highest point, of the parabola.
Identifying a Vertex
PracticeIdentify the vertex.
Practice
Vocabulary
■ You can use the fact that a parabola is symmetric to graph it quickly.
■ First, find the coordinates of the vertex and several points on one side of the
vertex.
■ Then reflect the points across the axis of symmetry.
■ For graphs of functions of the form y = 𝑎𝑥2, the vertex is at the origin.
■ The axis of symmetry is the y–axis, or x = 0.
Graphing 𝑦 = 𝑎𝑥2
Graph the function. Make a table of values. What are the domain and range?
1. 𝑦 =1
3𝑥2
2. 𝑦 = −3𝑥2
3. 𝑦 = 4𝑥2
Practice
Graph each function. Then identify the domain and range of the function.
1. 𝑦 = −4𝑥2
2. 𝑦 = −1
3𝑥2
3. 𝑓 𝑥 = 1.5𝑥2
4. 𝑓 𝑥 =2
3𝑥2
Vocabulary
■ The y – axis is the axis of symmetry for graphs of functions
of the form 𝑦 = 𝑎𝑥2 + 𝑐.
■ The value of c translates the graph up or down.
Graphing 𝑦 = 𝑎𝑥2 + 𝑐
What is the relationship of the following graphs?
1. 𝑦 = 2𝑥2 + 3 𝑎𝑛𝑑 𝑦 = 2𝑥2
2. 𝑦 = 𝑥2 𝑎𝑛𝑑 𝑦 = 𝑥2 − 3
3. 𝑦 = −1
2𝑥2 𝑎𝑛𝑑 𝑦 = −
1
2𝑥2 + 1
Practice
Graph each function.
1. 𝑓 𝑥 = 𝑥2 + 4
2. 𝑓 𝑥 = −𝑥2 − 3
3. 𝑓 𝑥 =1
2𝑥2 + 2
Vocabulary
As an object falls, its speed continues to increase, so its height above the ground decreases at a faster and faster rate.
Ignoring air resistance, you can model the object’s height with the function h = –16t2 + c.
The height h is in feet, the time t is in seconds, and the object’s initial height c is in feet.
Using the Falling Object Model
An acorn drops from a tree branch 20 feet above the ground. The function h = –16t2 + 20 gives the
height h of the acorn (in feet) after t seconds. What is the graph of this quadratic function? At what
time does the the acorn hit the ground?
t h = –16t2 + 20
Using the Falling Object Model
An acorn drops from a tree branch 70 feet above the ground. The function h = –16t2 + 70 gives the
height h of the acorn (in feet) after t seconds. What is the graph of this quadratic function? At what
time does the the acorn hit the ground?
t h = –16t2 + 70
PracticeA person walking across a bridge accidentally drops an orange into the
river below from a height of 40 feet. The function ℎ = −16𝑡2 + 40 gives
the orange’s approximate height h above the water, in feet, after t
seconds. In how many seconds will the orange hit the water?
A bird drops a stick to the ground from a height of 80 feet. The function
ℎ = −16𝑡2 + 80 gives the stick’s approximate height h above the
ground, in feet, after t seconds. Graph the function. At about what time
does the stick hit the ground?
QUADRATIC FUNCTIONS
Vocabulary■ In the quadratic function y = ax2 + bx + c, the value of b affects the position of the
axis of symmetry.
■ The axis of symmetry changes with each equation because of the change in the b-
value. The equation of the axis of symmetry is related to the ratio 𝑏
𝑎.
■ The equation of the axis of symmetry is 𝑥 = −1
2
𝑏
𝑎𝑜𝑟 𝑥 =
−𝑏
2𝑎.
■ Graph of a Quadratic Function
o The graph of y = ax2 + bx + c, where a ≠ 0, has the line 𝑥 =−𝑏
2𝑎as its axis of symmetry. The x–
coordinate of the vertex is −𝑏
2𝑎.
Vocabulary
■ When you substitute x = 0 into the equation y = ax2 + bx +
c, you get y = c. So the y–intercept of a quadratic function is
c.
■ You can use the axis of symmetry and the y–intercept to
help you graph a quadratic function.
Graphing y = ax2 + bx + c
What is the graph of the function? Show the axis of symmetry.
1. 𝑦 = 𝑥2 − 6𝑥 + 4
2. 𝑦 = −𝑥2 + 4𝑥 − 2
3. 𝑦 = 2𝑥2 + 3
4. 𝑦 = −3𝑥2 + 12𝑥 + 1
5. 𝑓 𝑥 = 𝑥2 + 4𝑥 − 5
6. 𝑓 𝑥 = −4𝑥2 + 11
Practice
What is the graph of the function? Show the line of symmetry.
1. 𝑦 = 2𝑥2 − 6𝑥 + 1
2. 𝑓 𝑥 = 2𝑥2 + 4𝑥 − 1
3. 𝑦 = 6𝑥2 + 6𝑥 − 5
4. 𝑓 𝑥 = −5𝑥2 + 3𝑥 + 2
5. 𝑦 = −2𝑥2 − 10𝑥
6. 𝑦 = −4𝑥2 − 16𝑥 − 3
Vocabulary
■ You have used h = –16t2 + c to find the height h above the ground
of an object falling from an initial height c at time t.
■ If an object projected into the air given an initial upward velocity v
continues with no additional force of its own, the formula h = –16t2
+ vt + c givens its approximate height above the ground.
Using a Vertical Motion Model
During halftime of a basketball game, a sling shot launches T–shirts at the crowd. A T–shirt launched with an initial upward velocity of 72 feet per second. The T–shirt is caught 35 feet above the court. The T–shirt is launched from a height of 5 feet.
a. How long will it take the T–shirt to reach its maximum height?
b. What is the maximum height?
c. What is the range of the function that models the height of the T–shirt over time?
Using a Vertical Motion Model
During halftime of a basketball game, a sling shot launches T–shirts at the crowd. A T–shirt launched with an initial upward velocity of 64 feet per second. The T–shirt is caught 35 feet above the court. The T–shirt is launched from a height of 5 feet.
a. How long will it take the T–shirt to reach its maximum height?
b. What is the maximum height?
c. What is the range of the function that models the height of the T–shirt over time?
Practice
A baseball is thrown into the air with an upward velocity of 30 feet per second. Its
height h, in feet, after t seconds is given by the function h = –16t2 + 30t + 6.
a. How long will it take the ball to reach its maximum height?
b. What is the ball’s maximum height?
c. What is the range of the function?